I tried to de-activate the Ajax Search Pro and then tried the log-in pop-up to log-in, but it's still not working – which means that the issue is very likely not related to this plugin.

What I can see on the console is a 401 error whenever an ajax request is sent to the WordPress admin-ajax.php handler, when the login button is clicked: https://i.imgur.com/IzzDnaj.png
This could indicate an invalid .htaccess rule that blocks the file, or a .htpassword protection on that folder, maybe an permission issue, or some way of protection over on that file. Perhaps this may help you to get it resolved.

Best,Ernest MarcinkoIf you like my products, don't forget to rate them on codecanyon :)